A Neural Network-based Approach to Determining Vesicle Shapes in a Phase Field Model

We introduce a novel framework to numerically solve the energetically optimized shape of a vesicle by a feed-forward neural network. This framework is akin to the variational formalism for minimizing the elastic bending energy of vesicle membranes subject to volume and area constraints, which was first formulated by Helfrich and was investigated further by Seifert et. al. We model the vesicles using a phase field approach proposed by Wang et. al. Following this approach, the inputs of the network are the coordinates of a point inside a three-dimensional domain of interest and the output is the value of the corresponding phase field. As such, the neural network corresponds to an ansatz for an equivalent energy-based variational problem, which converges to the solution by minimizing a loss function for the elastic bending energy with appropriate volume and surface area penalty terms. The three main axisymmetric shapes i.e., prolate or dumbbell, oblate or discocyte and stomatocyte, are obtained. Moreover, nonaxisymmetric shapes and shapes that correspond to a non-zero spontaneous curvature parameter in Helfrich’s energy equation are attainable by implementing this framework.

We acknowledge funding support from the Natural Sciences and Engineering Research Council of Canada.